**Johann Bernoulli** was a Swiss mathematician who studied reflection and refraction of light, orthogonal trajectories of families of curves, quadrature of areas by series and the brachistochrone.

- Johann's first publication was on the process of fermentation in 1690, certainly not a mathematical topic but in 1691 Johann went to Geneva where he lectured on the differential calculus.
- From Geneva, Johann made his way to Paris and there he met mathematicians in Malebranche's circle, where the focus of French mathematics was at that time.
- There Johann met de l'Hôpital and they engaged in deep mathematical conversations.
- Contrary to what is commonly said these days, de l'Hôpital was a fine mathematician, perhaps the best mathematician in Paris at that time, although he was not quite in the same class as Johann Bernoulli.
- De l'Hôpital was delighted to discover that Johann Bernoulli understood the new calculus methods that Leibniz had just published and he asked Johann to teach him these methods.
- This Johann agreed to do and the lessons were taught both in Paris and also at de l'Hôpital's country house at Oucques.
- Bernoulli received generous payment from de l'Hôpital for these lessons, and indeed they were worth a lot for few other people would have been able to have given them.
- After Bernoulli returned to Basel he still continued his calculus lessons by correspondence, and this did not come cheap for de l'Hôpital who paid Bernoulli half a professor's salary for the instruction.
- However it did assure de l'Hôpital of a place in the history of mathematics since he published the first calculus book Analyse des infiniment petits pour l'intelligence des lignes courbes Ⓣ(Infinitesimal analysis for understanding curved lines) (1696) which was based on the lessons that Johann Bernoulli sent to him.
- As one would expect, it upset Johann Bernoulli greatly that this work did not acknowledge the fact that it was based on his lectures.
- The well known de l'Hôpital's rule is contained in this calculus book and it is therefore a result of Johann Bernoulli.
- Proof that the work was due to Bernoulli was not obtained until 1922 when a copy of Johann Bernoulli's course made by his nephew Nicolaus(I) Bernoulli was found in Basel.
- Bernoulli's course is virtually identical with de l'Hôpital's book but it is worth pointing out that de l'Hôpital had corrected a number of errors such as Bernoulli's mistaken belief that the integral of 1/x1/x1/x is finite.
- After de l'Hôpital's death in 1704 Bernoulli protested strongly that he was the author of de l'Hôpital's calculus book.
- It appears that the handsome payment de l'Hôpital made to Bernoulli carried with it conditions which prevented him speaking out earlier.
- However, few believed Johann Bernoulli until the proofs discovered in 1922.
- Let us return to an account of Bernoulli's time in Paris.
- In 1692, while in Paris, he met Varignon and this resulted in a strong friendship and also Varignon learned much about applications of the calculus from Johann Bernoulli over the many years which they corresponded.
- Johann Bernoulli also began a correspondence with Leibniz which was to prove very fruitful.
- This was a period of considerable mathematical achievement for Johann Bernoulli.
- At this stage Johann and Jacob were learning much from each other in a reasonably friendly rivalry which, a few years later, would descend into open hostility.
- We mentioned above that Johann's doctoral dissertation was on a topic in medicine, but it was really on an application of mathematics to medicine, being on muscular movement, and it was submitted in 1694.
- Johann did not wish to follow a career in medicine however, but there were little prospects of a chair at Basel in mathematics since Jacob filled this post.
- A stream of mathematical ideas continued to flow from Johann Bernoulli.
- Integration to Bernoulli was simply viewed as the inverse operation to differentiation and with this approach he had great success in integrating differential equations.
- The fault was not all on Jacob's side however, and Johann was equally to blame for the deteriorating relations.
- It is interesting to note that Johann was appointed to the chair of mathematics but his letter of appointment mentions his medical skills and offered him the chance to practise medicine while in Groningen.
- In a second dispute in 1702 Bernoulli was accused by a student at the University of Groningen, Petrus Venhuysen, who published a pamphlet which basically accused Bernoulli of following Descartes' philosophy.
- This was not Johann's only dispute while in Groningen.
- Johann proposed the problem of the brachristochrone in June 1696 and challenged others to solve it.
- Five solutions were obtained, Jacob Bernoulli and Leibniz both solving the problem in addition to Johann Bernoulli.
- Johann's solution to this problem was less satisfactory than that of Jacob but, when Johann returned to the problem in 1718 having read a work by Taylor, he produced an elegant solution which was to form a foundation for the calculus of variations.
- Hence Johann was not returning to Basel expecting the chair of mathematics, rather he was returning to fill the chair of Greek.
- Before reaching Basel, however, Johann was tempted by an offer of a chair at the University of Utrecht.
- The head of the University of Utrecht was so keen to have Bernoulli come there that he set out after the Bernoulli's catching up with them in Frankfurt.
- He tried to persuade Johann to go to Utrecht but Bernoulli was set on returning to Basel.
- There were other offers that Johann turned down, such as Leiden, a second offer from Utrecht and a generous offer for him to return to Groningen in 1717.
- In 1713 Johann became involved in the Newton-Leibniz controversy.
- Although Bernoulli was essentially correct in his support of the superior calculus methods of Leibniz, he also supported Descartes' vortex theory over Newton's theory of gravitation and here he was certainly incorrect.
- Bernoulli also made important contributions to mechanics with his work on kinetic energy, which, not surprisingly, was another topic on which mathematicians argued over for many years.
- Daniel Bernoulli completed his most important work Hydrodynamica Ⓣ(Hydrodynamics) in 1734 and published it in 1738 at about the same time as Johann published Hydraulica.
- Johann Bernoulli attained great fame in his lifetime.

Born 27 July 1667, Basel, Switzerland. Died 1 January 1748, Basel, Switzerland.

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**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive